In $\triangle ABC$,the points $P, Q, R$ divide $BC, CA, AB$ in the ratio $3:4, 2:5, 9:5$,respectively,and the point $D$ divides $BC$ in the ratio $2:3$. If $\vec{AP} + \vec{BQ} + \vec{CR} = k \vec{AD}$,then $(14k + 1) : (14k - 1) = $

Let $\overline{a}=3 \hat{i}-\alpha \hat{j}+\hat{k}$ and $\overline{b}=\hat{i}+\alpha \hat{j}+3 \hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\overline{a}$ and $\overline{b}$ is $8 \sqrt{3}$ sq. units,then $\overline{a} \cdot \overline{b}$ is equal to

If the angle between two vectors $\vec{u} = i + k$ and $\vec{v} = i - j + ak$ is $\pi / 3,$ then the value of $a$ is:

Let the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ be such that $|\overrightarrow{a}|=2, |\overrightarrow{b}|=4$ and $|\overrightarrow{c}|=4$. If the projection of $\overrightarrow{b}$ on $\overrightarrow{a}$ is equal to the projection of $\overrightarrow{c}$ on $\overrightarrow{a}$ and $\overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$,then the value of $|\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}|$ is

If two vectors $\vec{a}$ and $\vec{b}$ which are perpendicular to each other are such that $|\vec{a}|=8$ and $|\vec{b}|=3$,then $|\vec{a}-2\vec{b}|=$

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be vectors of equal magnitude such that the angle between $\vec{a}$ and $\vec{b}$ is $\alpha$,$\vec{b}$ and $\vec{c}$ is $\beta$,and $\vec{c}$ and $\vec{a}$ is $\gamma$. Then the minimum value of $\cos \alpha + \cos \beta + \cos \gamma$ is